3.1.25 \(\int \frac {1}{(5-3 \cos (c+d x))^4} \, dx\) [25]

Optimal. Leaf size=108 \[ \frac {385 x}{32768}+\frac {385 \text {ArcTan}\left (\frac {\sin (c+d x)}{3-\cos (c+d x)}\right )}{16384 d}+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac {311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))} \]

[Out]

385/32768*x+385/16384*arctan(sin(d*x+c)/(3-cos(d*x+c)))/d+1/16*sin(d*x+c)/d/(5-3*cos(d*x+c))^3+25/512*sin(d*x+
c)/d/(5-3*cos(d*x+c))^2+311/8192*sin(d*x+c)/d/(5-3*cos(d*x+c))

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Rubi [A]
time = 0.07, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2743, 2833, 12, 2736} \begin {gather*} \frac {385 \text {ArcTan}\left (\frac {\sin (c+d x)}{3-\cos (c+d x)}\right )}{16384 d}+\frac {311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}+\frac {25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac {385 x}{32768} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - 3*Cos[c + d*x])^(-4),x]

[Out]

(385*x)/32768 + (385*ArcTan[Sin[c + d*x]/(3 - Cos[c + d*x])])/(16384*d) + Sin[c + d*x]/(16*d*(5 - 3*Cos[c + d*
x])^3) + (25*Sin[c + d*x])/(512*d*(5 - 3*Cos[c + d*x])^2) + (311*Sin[c + d*x])/(8192*d*(5 - 3*Cos[c + d*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2736

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2/(d*q))*ArcTan[b*(Cos[c + d*x]/(a + q + b*Sin[c + d*x]))], x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && PosQ[a]

Rule 2743

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
+ 1)/(d*(n + 1)*(a^2 - b^2))), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n +
1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ
erQ[2*n]

Rule 2833

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(
b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(a^2 - b
^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rubi steps

\begin {align*} \int \frac {1}{(5-3 \cos (c+d x))^4} \, dx &=\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}-\frac {1}{48} \int \frac {-15-6 \cos (c+d x)}{(5-3 \cos (c+d x))^3} \, dx\\ &=\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac {\int \frac {186+75 \cos (c+d x)}{(5-3 \cos (c+d x))^2} \, dx}{1536}\\ &=\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac {311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}-\frac {\int -\frac {1155}{5-3 \cos (c+d x)} \, dx}{24576}\\ &=\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac {311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}+\frac {385 \int \frac {1}{5-3 \cos (c+d x)} \, dx}{8192}\\ &=\frac {385 x}{32768}+\frac {385 \tan ^{-1}\left (\frac {\sin (c+d x)}{3-\cos (c+d x)}\right )}{16384 d}+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac {311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 66, normalized size = 0.61 \begin {gather*} \frac {770 \text {ArcTan}\left (2 \tan \left (\frac {1}{2} (c+d x)\right )\right )-\frac {9 (4883 \sin (c+d x)-2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(-5+3 \cos (c+d x))^3}}{32768 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - 3*Cos[c + d*x])^(-4),x]

[Out]

(770*ArcTan[2*Tan[(c + d*x)/2]] - (9*(4883*Sin[c + d*x] - 2340*Sin[2*(c + d*x)] + 311*Sin[3*(c + d*x)]))/(-5 +
3*Cos[c + d*x])^3)/(32768*d)

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Maple [A]
time = 0.10, size = 77, normalized size = 0.71

method	result	size
derivativedivides	\(\frac {\frac {\frac {369 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {639 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}}{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right )^{3}}+\frac {385 \arctan \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}}{d}\)	\(77\)
default	\(\frac {\frac {\frac {369 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {639 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}}{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right )^{3}}+\frac {385 \arctan \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}}{d}\)	\(77\)
risch	\(\frac {i \left (10395 \,{\mathrm e}^{5 i \left (d x +c \right )}-86625 \,{\mathrm e}^{4 i \left (d x +c \right )}+239470 \,{\mathrm e}^{3 i \left (d x +c \right )}-218466 \,{\mathrm e}^{2 i \left (d x +c \right )}+73575 \,{\mathrm e}^{i \left (d x +c \right )}-8397\right )}{12288 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-10 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )^{3}}+\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-3\right )}{32768 d}-\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {1}{3}\right )}{32768 d}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5-3*cos(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(8*(369/4096*tan(1/2*d*x+1/2*c)^5+117/2048*tan(1/2*d*x+1/2*c)^3+639/65536*tan(1/2*d*x+1/2*c))/(4*tan(1/2*d
*x+1/2*c)^2+1)^3+385/16384*arctan(2*tan(1/2*d*x+1/2*c)))

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Maxima [A]
time = 0.49, size = 152, normalized size = 1.41 \begin {gather*} \frac {\frac {18 \, {\left (\frac {71 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {416 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {656 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {64 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1} + 385 \, \arctan \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{16384 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5-3*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

1/16384*(18*(71*sin(d*x + c)/(cos(d*x + c) + 1) + 416*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 656*sin(d*x + c)^5
/(cos(d*x + c) + 1)^5)/(12*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 48*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 64*s
in(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1) + 385*arctan(2*sin(d*x + c)/(cos(d*x + c) + 1)))/d

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Fricas [A]
time = 0.37, size = 121, normalized size = 1.12 \begin {gather*} -\frac {385 \, {\left (27 \, \cos \left (d x + c\right )^{3} - 135 \, \cos \left (d x + c\right )^{2} + 225 \, \cos \left (d x + c\right ) - 125\right )} \arctan \left (\frac {5 \, \cos \left (d x + c\right ) - 3}{4 \, \sin \left (d x + c\right )}\right ) + 36 \, {\left (311 \, \cos \left (d x + c\right )^{2} - 1170 \, \cos \left (d x + c\right ) + 1143\right )} \sin \left (d x + c\right )}{32768 \, {\left (27 \, d \cos \left (d x + c\right )^{3} - 135 \, d \cos \left (d x + c\right )^{2} + 225 \, d \cos \left (d x + c\right ) - 125 \, d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5-3*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/32768*(385*(27*cos(d*x + c)^3 - 135*cos(d*x + c)^2 + 225*cos(d*x + c) - 125)*arctan(1/4*(5*cos(d*x + c) - 3
)/sin(d*x + c)) + 36*(311*cos(d*x + c)^2 - 1170*cos(d*x + c) + 1143)*sin(d*x + c))/(27*d*cos(d*x + c)^3 - 135*
d*cos(d*x + c)^2 + 225*d*cos(d*x + c) - 125*d)

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Sympy [C] Result contains complex when optimal does not.
time = 3.17, size = 597, normalized size = 5.53 \begin {gather*} \begin {cases} \frac {x}{\left (5 - 3 \cosh {\left (2 \operatorname {atanh}{\left (\frac {1}{2} \right )} \right )}\right )^{4}} & \text {for}\: c = - d x - 2 i \operatorname {atanh}{\left (\frac {1}{2} \right )} \vee c = - d x + 2 i \operatorname {atanh}{\left (\frac {1}{2} \right )} \\\frac {x}{\left (5 - 3 \cos {\left (c \right )}\right )^{4}} & \text {for}\: d = 0 \\\frac {24640 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {18480 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {4620 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {385 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {11808 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {7488 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {1278 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5-3*cos(d*x+c))**4,x)

[Out]

Piecewise((x/(5 - 3*cosh(2*atanh(1/2)))**4, Eq(c, -d*x - 2*I*atanh(1/2)) | Eq(c, -d*x + 2*I*atanh(1/2))), (x/(
5 - 3*cos(c))**4, Eq(d, 0)), (24640*(atan(2*tan(c/2 + d*x/2)) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d
*x/2)**6/(1048576*d*tan(c/2 + d*x/2)**6 + 786432*d*tan(c/2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16384*
d) + 18480*(atan(2*tan(c/2 + d*x/2)) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**4/(1048576*d*tan(c
/2 + d*x/2)**6 + 786432*d*tan(c/2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16384*d) + 4620*(atan(2*tan(c/2
+ d*x/2)) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**2/(1048576*d*tan(c/2 + d*x/2)**6 + 786432*d*
tan(c/2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16384*d) + 385*(atan(2*tan(c/2 + d*x/2)) + pi*floor((c/2
+ d*x/2 - pi/2)/pi))/(1048576*d*tan(c/2 + d*x/2)**6 + 786432*d*tan(c/2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)
**2 + 16384*d) + 11808*tan(c/2 + d*x/2)**5/(1048576*d*tan(c/2 + d*x/2)**6 + 786432*d*tan(c/2 + d*x/2)**4 + 196
608*d*tan(c/2 + d*x/2)**2 + 16384*d) + 7488*tan(c/2 + d*x/2)**3/(1048576*d*tan(c/2 + d*x/2)**6 + 786432*d*tan(
c/2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16384*d) + 1278*tan(c/2 + d*x/2)/(1048576*d*tan(c/2 + d*x/2)*
*6 + 786432*d*tan(c/2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16384*d), True))

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Giac [A]
time = 0.49, size = 90, normalized size = 0.83 \begin {gather*} \frac {385 \, d x + 385 \, c + \frac {36 \, {\left (656 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 416 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 71 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}} - 770 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) - 3}\right )}{32768 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5-3*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/32768*(385*d*x + 385*c + 36*(656*tan(1/2*d*x + 1/2*c)^5 + 416*tan(1/2*d*x + 1/2*c)^3 + 71*tan(1/2*d*x + 1/2*
c))/(4*tan(1/2*d*x + 1/2*c)^2 + 1)^3 - 770*arctan(sin(d*x + c)/(cos(d*x + c) - 3)))/d

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Mupad [B]
time = 0.49, size = 97, normalized size = 0.90 \begin {gather*} \frac {385\,\mathrm {atan}\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16384\,d}-\frac {385\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{16384\,d}+\frac {\frac {369\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{512}+\frac {117\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{256}+\frac {639\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8192}}{d\,{\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*cos(c + d*x) - 5)^4,x)

[Out]

(385*atan(2*tan(c/2 + (d*x)/2)))/(16384*d) - (385*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(16384*d) + ((639*tan(
c/2 + (d*x)/2))/8192 + (117*tan(c/2 + (d*x)/2)^3)/256 + (369*tan(c/2 + (d*x)/2)^5)/512)/(d*(4*tan(c/2 + (d*x)/
2)^2 + 1)^3)

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